Examples and Applications of Noncommutative Geometry and K-Theory

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Examples and Applications of Noncommutative Geometry and K-Theory EXAMPLES AND APPLICATIONS OF NONCOMMUTATIVE GEOMETRY AND K-THEORY JONATHAN ROSENBERG Abstract. These are informal notes from my course at the 3era Escuela de Invierno Luis Santal´o-CIMPA Research School on Topics in Noncommutative Geometry. Feedback, especially from participants at the course, is very wel- come. The course basically is divided into two (related) sections. Lectures 1–3 deal with Kasparov’s KK-theory and some of its applications. Lectures 4–5 deal with one of the most fundamental examples in noncommutative geometry, the noncommuative 2-torus. Contents Lecture 1. Introduction to Kasparov’s KK-theory 3 1.1. Why KK? 3 1.2. Kasparov’s original definition 4 1.3. Connections and the product 7 1.4. Cuntz’s approach 8 1.5. Higson’s approach 9 Lecture 2. K-theory and KK-theoryofcrossedproducts 11 2.1. Equivariant Kasparov theory 11 2.2. Basicpropertiesofcrossedproducts 12 2.3. The dual action and Takai duality 13 2.4. Connes’ “Thom isomorphism” 13 2.5. The Pimsner-Voiculescu Theorem 16 2.6. The Baum-Connes Conjecture 16 2.7. The approach of Meyer and Nest 18 Lecture 3. The universal coefficient theorem for KK and some of its applications 19 3.1. Introduction to the UCT 19 3.2. TheproofofRosenbergandSchochet 19 3.3. The equivariant case 20 3.4. The categorical approach 20 3.5. The homotopy-theoretic approach 22 3.6. Topological applications 22 3.7. Applications to C∗-algebras 23 Date: 2010. 2010 Mathematics Subject Classification. Primary 46L85; Secondary 19-02 46L87 19K35. Preparation of these lectures was partially supported by NSF grant DMS-0805003 and DMS- 1041576. The author thanks the members of the GNC Group at Buenos Aires, and especially Guillermo Corti˜nas, for their excellent organization of the Winter School. 1 2 JONATHAN ROSENBERG Lecture 4. A fundamental example in noncommutative geometry: topology and geometry of the irrational rotation algebra 24 4.1. Basic facts about Aθ 24 ∞ 4.2. Basic facts about Aθ 26 4.3. Geometry of vector bundles 27 4.4. Miscellaneous other facts about Aθ 28 Lecture 5. Applications of the irrational rotation algebra in number theory and physics 29 5.1. Applications to number theory 29 5.2. Applications to physics 29 References 30 APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 3 Lecture 1. Introduction to Kasparov’s KK-theory 1.1. Why KK? KK-theory is a bivariant version of topological K-theory, defined for C∗-algebras, with or without a group action. It can be defined for either real or complex algebras, but in these notes we will stick to complex algebras for sim- plicity. Thus if A and B are complex C∗-algebras, subject to a minor technical requirement (that B be σ-unital, which is certainly the case if it is either uni- tal or separable), an abelian group KK(A, B) is defined, with the property that KK(C,B)= K(B)= K0(B) if the first algebra A is just the scalars. (For the basic properties of K0, I refer you to the courses by Reich and Karoubi.) The theory was defined by Gennadi Kasparov in a remarkable series of papers: [33, 34, 35]. However, the definition at first seems highly technical and unmotivated, so it’s worth first seeing where the theory comes from and why one might be interested in it. For purposes of this introduction, we will only be concerned with the case where A and B are commutative. Thus A = C0(X) and B = C0(Y ), where X and Y are locally compact Hausdorff spaces. We will abbreviate KK(C0(X), C0(Y )) to KK(X, Y ). It is worth pointing out that the study of KK(X, Y ) (without considering KK(A, B) more generally) is already highly nontrivial, and encom- passes most of the features of the general theory. Note that we expect to have KK(C, C0(Y )) = KK(pt, Y ) = K(Y ), the K-theory of Y with compact support. Recall that this is the Grothendieck group of complexes of vector bundles that are exact off a compact set. It’s actually enough to take complexes of length 2, so an element of K(Y ) is represented by a pair of vector bundles V and V ′ over Y , ϕ together with a morphism of vector bundles V −→ V ′ that is an isomorphism off a compact set. Alternatively, K(Y ) can be identified with the reduced K-theory K(Y+) of the one-point compactification Y+ of Y . A good place to start in trying to understand KK is Atiyah’s paper [4] on thee Bott periodicity theorem. Bott periodicity, or more generally, the Thom iso- morphism theorem for a complex vector bundle, asserts that if p: E → X is a complex vector bundle (more generally, one could take an even-dimensional real vector bundle with a spinc structure), then there is a natural isomorphism βE : K(X) → K(E), called the Thom isomorphism in K-theory. In the special case where X = pt, E is just Cn for some n, and we are asserting that there is a natural isomorphism Z = K(pt) → K(Cn) = K(R2n) = K(S2n), the Bott pe- ∗ riodicity map. The map βE can be described by the formula βE(a) = p (a) · τE. Here p∗(a) is the pull-back of a ∈ K(X) to E. Since a had compacte support, p∗(a) has compact support in the base direction of E, but is constant on fibers of p, so it certainly does not have compact support in the fiber direction. However, we can multiply it by the Thom class τE, which does have compact support along the fibers, and the product will have compact support in both directions, and will thus give a class in K(E) (remember that since E is necessarily noncompact, assuming n > 0, we need to use K-theory with compact support). The Thom class τE, in turn, can be described [59, §3] as an explicit complex • p∗E over E. The vector bundles in this complex are the exterior powers of E pulled back from X to E, and j j+1 V the map at a point e ∈ Ex from Ex to Ex is simply exterior product with e. This complex has compact support in the fiber directions since it is exact off the zero-section of E. (If e 6= 0, thenV the kernelV of e ∧ is spanned by products e ∧ ω.) 4 JONATHAN ROSENBERG So far this is all simple vector bundle theory and KK is not needed. But it comes in at the next step. How do we prove that βE is an isomorphism? The simplest way would be to construct an inverse map αE : K(E) → K(X). But there is no easy way to describe such a map using topology alone. As Atiyah recognized, the easiest way to construct αE uses elliptic operators, in fact the family of Dolbeault operators along the fibers of E. Thus whether we like it or not, some analysis comes in at this stage. In more modern language, what we really want is the class αE in KK(E,X) corresponding to this family of operators, and the verification of the Thom isomorphism theorem is a Kasparov product calculation, the fact that αE is a KK inverse to the class βE ∈ KK(X, E) described (in slightly other terms) before. Atiyah also noticed [4] that it’s really just enough (because of certain identities about products) to prove that αE is a one-way inverse to βE, or in other words, in the language of Kasparov theory, that βE ⊗E αE = 1X . This comes down to an index calculation, which because of naturality comes down to the single calculation β ⊗C α = 1 ∈ KK(pt, pt) when X is a point and E = C, which amounts to the Riemann-Roch theorem for CP1. What then is KK(X, Y ) when X and Y are locally compact spaces? An element of KK(X, Y ) defines a map of K-groups K(X) → K(Y ), but is more than this; it is in effect a natural family of maps of K-groups K(X × Z) → K(Y × Z) for arbitrary Z. Naturality of course means that one gets a natural transformation of functors, from Z 7→ K(X × Z) to Z 7→ K(Y × Z). (Nigel Higson has pointed out that one can use this in reverse to define KK(X, Y ) as a natural family of maps of K-groups K(X × Z) → K(Y × Z) for arbitrary Z. The reason why this works will be explained in Lecture 3 in this series.) In particular, since KK(X × Rj ) can be identified with K−j(X), an element of KK(X, Y ) defines a graded map of K-groups Kj(X) → Kj(Y ), at least for j ≤ 0 (but then for arbitrary j because of Bott periodicity). The example of Atiyah’s class αE ∈ KK(E,X), based on a family of elliptic operators over E parametrized by X, shows that one gets an element of the bivariant K-group KK(X, Y ) from a family of elliptic operators over X parametrized by Y . The element that one gets should be invariant under homotopies of such operators. Hence Kasparov’s definition of KK(A, B) is based on a notion of homotopy classes of generalized elliptic operators for the first algebra A, “parametrized” by the second algebra B (and thus commuting with a B-module structure). 1.2. Kasparov’s original definition. As indicated above in Section 1.1, an ele- ment of KK(A, B) is roughly speaking supposed to be a homotopy class of families of elliptic pseudodifferential operators over A parametrized by B.
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